Finite Group Databasebeta

← #10695568

Subgroups — #10695568

Order 768.

768 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78624C211yesmaximal, characteristic, complemented
2#74129C211yesmaximal, characteristic, complemented
3#73597C211yesmaximal, characteristic
4#69967C211yesmaximal, characteristic, complemented
5#69531C211yesmaximal, characteristic, complemented
6#69442C211yesmaximal, characteristic, complemented
7C3×#1284C211yesmaximal, characteristic, complemented
8#9940—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C2×#886—21no—
13C2×#882—21no—
14C3×#407C2211yescharacteristic
15C3×#381C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5207C2211yescharacteristic
18#5131—21no—
19#5121—21no—
20#4955—21no—
21#4950—21no—
22#4939—22no—
23#4737—22no—
24#4736—21no—
25#4735—21no—
26#4677—21no—
27#4677—21no—
28#4674—21no—
29#4671—21no—
30#4670—22no—
31#4523—22no—
32#4468—22no—
33#4451—22no—
34#3094—31no—
35#2539—31no—
36#2482—31no—
37#1826—31no—
38#1510—31no—
39#1462—31no—
40#1284D311yescharacteristic, complemented
41C24×D3D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C23×Dic3D411yescharacteristic
45C22×D12—22no—
46C22×D12D411yescharacteristic
47C2×C3×M16—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42D412yes—
51C2×#217—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67#926—21no—
68#926—21no—
69#926—21no—
70#926—21no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3D411yescharacteristic, complemented
75C2×C4⋊Dic3D411yescharacteristic, complemented
76C2×C4×Dic3—21no—
77C2×#208—21no—
78#886—44no—
79#882—44no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yescomplemented
82C3×#95—21no—
83C3×#95—21no—
84C3×#95D412yescomplemented
85C3⋊#95—22no—
86#830—21no—
87#830—21no—
88#821—22no—
89#821—22no—
90#818—22no—
91#817—22no—
92C2×#123—61no—
93C2×#121—31no—
94C2×#120—61no—
95C2×#117—31no—
96C22×C22⋊C4—31no—
97#482—62no—
98#481—61no—
99#480—61no—
100#428—61no—
101#418—61no—
102#407—31no—
103#407C2×D311yescharacteristic
104#399—61no—
105#399—61no—
106#397—61no—
107#392—61no—
108#390—62no—
109#388—62no—
110#385—61no—
111#381C2×D311yescharacteristic
112#379—61no—
113C4×C22⋊C4C2×D311yescharacteristic
114#339—62no—
115#327—62no—
116#323—62no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C23×D3—21no—
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—42no—
123C23×D3—42no—
124C23×D3—42no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—42no—
135C2×#67—42no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—42no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C2×D12—44no—
149C2×D12—44no—
150C2×D12—82no—
151C3×M16—42no—
152C2×C3×C8—22no—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—42no—
155C3×C42—42no—
156#217—41no—
157#217—41no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—82no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—44no—
175C4⋊Dic3—82no—
176C4×Dic3—41no—
177C4×Dic3—41no—
178#208—42no—
179C2×C3⋊C8—41no—
180C25—31no—
181C22×D4—31no—
182C22×D4—62no—
183C22×D4—61no—
184C23×C4—31no—
185C23×C4C22×D311yescharacteristic
186C2×M16—21no—
187C2×M16—61no—
188#123—124no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193#120—124no—
194#117—61no—
195#117—61no—
196#117—61no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C42—61no—
218C2×C42D1212yes—
219#107—62no—
220#106—62no—
221#102—62no—
222#102—62no—
223C22⋊C8—21no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8D1212yescomplemented
227#95—21no—
228#95—21no—
229#95—62no—
230#95D1212yescomplemented
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3#12011yescharacteristic
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C22×D3—22no—
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—42no—
253C22×D3—88no—
254C22×D3—81no—
255C22×D3—81no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12711yescharacteristic
272#67—82no—
273#67—82no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277C2×Dic3—42no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—81no—
285D12—42no—
286D12—81no—
287D12—88no—
288C3×C8—42no—
289C3⋊C8—81no—
290C24—61no—
291C24—61no—
292C24—61no—
293C24—61no—
294C24—122no—
295C24—122no—
296C24—122no—
297C24D3×D411yescharacteristic
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—122no—
301C2×D4—124no—
302C2×D4—124no—
303C2×D4—122no—
304C2×D4—122no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—242no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—122no—
318C22×C4C2×D1211yescharacteristic
319C22×C4C2×D1211yescharacteristic
320C22×C4C2×D1211yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4#23911yescharacteristic
324M16—42no—
325M16—122no—
326C2×C8—22no—
327C2×C8—121no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—124no—
335C4⋊C4—242no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22⋊C4—124no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—121no—
348C22⋊C4—121no—
349C22⋊C4—242no—
350C42—42no—
351C42—121no—
352C42—121no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3#39211yescharacteristic
362C22×C3#45312yes—
363C22×C3#45812yes—
364C2×D3—44no—
365C2×D3—44no—
366C2×D3—42no—
367C2×D3—82no—
368C2×D3—82no—
369C2×D3—82no—
370C2×D3—81no—
371C2×D3—81no—
372C2×D3—82no—
373C2×D3—82no—
374C2×D3—82no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—44no—
383C3×C4—82no—
384Dic3—81no—
385Dic3—81no—
386Dic3—81no—
387Dic3—81no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—62no—
392C23—62no—
393C23—62no—
394C23—122no—
395C23—121no—
396C23—121no—
397C23—122no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23—122no—
402C23—121no—
403C23—122no—
404C23—122no—
405C23—121no—
406C23—122no—
407C23—241no—
408C23—241no—
409C23—248no—
410C23#88211yescharacteristic
411C23#88211yescharacteristic
412C23#89511yescharacteristic
413D4—122no—
414D4—241no—
415D4—248no—
416D4—242no—
417D4—242no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—241no—
441C2×C4#87411yescharacteristic
442C2×C4#88211yescharacteristic
443C2×C4#89511yescharacteristic
444C2×C4#89511yescharacteristic
445C8—42no—
446C8—241no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3#200211yescharacteristic
451C2×C3#201011yescharacteristic
452C2×C3#202211yescharacteristic
453D3—82no—
454D3—161no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—124no—
464C22—124no—
465C22—122no—
466C22—242no—
467C22—242no—
468C22—242no—
469C22—242no—
470C22—242no—
471C22—242no—
472C22—241no—
473C22—241no—
474C22—241no—
475C22—241no—
476C22—241no—
477C22#467111yescharacteristic
478C22#474212yes—
479C22#482212yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—44no—
485C4—82no—
486C4—241no—
487C4—241no—
488C4—241no—
489C4—241no—
490C3#994011yescharacteristic, complemented, Sylow
491C2—22no—
492C2—22no—
493C2—41no—
494C2—242no—
495C2—481no—
496C2#6870311yescharacteristic
497C2#6876111yescharacteristic
498C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#392
prime radical#73597C2
hypercenter#1284D3
hypocenterC3#9940
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #78624C2semidirect
2. #74129C2semidirect
4. #69967C2semidirect
5. #69531C2semidirect
6. #69442C2semidirect
7. C3×#1284C2semidirect
40. #1284D3semidirect
74. C2×C4⋊Dic3D4semidirect
75. C2×C4⋊Dic3D4semidirect
81. C3×C22⋊C8D4semidirect
84. C3×#95D4semidirect
226. C22⋊C8D12semidirect
230. #95D12semidirect
490. C3#9940semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#786242
#741292
#735972
#699672
#695312
#694422
C3×#12842
#99403